This question tripped me up at first but then I figured out the correct answer. I thought it was a neat question if anyone wants to try it just for fun.
Given tan θ = 20/21 and 180° < θ < 270° , find the exact value of cos(θ/2) .
θ ≈ 223.6 .... so .... θ/2 will lie in the 2nd quadrant
r = √[ 20^2 + 21^2 ] = √841 = 29
cos (θ) = -21/ 29
cos (θ/2) = - √ [ ( 1 + ( -21 / 29 ) ) / 2 ]
cos (θ/2) = - √ [ (29 - 21) / (2*29) ] = - √ [ (8) / (2*29) ] = - √ [ 4 /29) ] =
- 2 / √ 29
Ah you got it!!
I got messed up because if you try to check it in WolframAlpha by entering
cos(atan(20/21)/2)
you get a different answer, but if you enter
cos((atan(20/21)+pi)/2)
it gives the right answer.